Question: You have found the following ages (in years) of all 4 sloths at your local zoo: $ 12,\enspace 12,\enspace 6,\enspace 1$ What is the average age of the sloths at your zoo? What is the variance? You may round your answers to the nearest tenth.
Because we have data for all 4 sloths at the zoo, we are able to calculate the population mean $({\mu})$ and population variance $({\sigma^2})$ To find the population mean , add up the values of all $4$ ages and divide by $4$ $ {\mu} = \dfrac{\sum\limits_{i=1}^{{N}} x_i}{{N}} = \dfrac{\sum\limits_{i=1}^{{4}} x_i}{{4}} $ $ {\mu} = \dfrac{12 + 12 + 6 + 1}{{4}} = {7.8\text{ years old}} $ Find the squared deviations from the mean for each sloth. Age $x_i$ Distance from the mean $(x_i - {\mu})$ $(x_i - {\mu})^2$ $12$ years $4.2$ years $17.64$ years $^2$ $12$ years $4.2$ years $17.64$ years $^2$ $6$ years $-1.8$ years $3.24$ years $^2$ $1$ year $-6.8$ years $46.24$ years $^2$ Because we used the population mean $({\mu})$ to compute the squared deviations from the mean , we can find the variance $({\sigma^2})$ , without introducing any bias, by simply averaging the squared deviations from the mean $ {\sigma^2} = \dfrac{\sum\limits_{i=1}^{{N}} (x_i - {\mu})^2}{{N}} $ $ {\sigma^2} = \dfrac{{17.64} + {17.64} + {3.24} + {46.24}} {{4}} $ $ {\sigma^2} = \dfrac{{84.76}}{{4}} = {21.19\text{ years}^2} $ The average sloth at the zoo is 7.8 years old. The population variance is 21.19 years $^2$.